246 XIV INTEGRATION IN LOCALLY COMPACT GROUPS We shall now show that (14.1.5.3) !(/+/') = !(/) + !(/') for all/,/' in Jf * . By (iii), it is enough to prove that!(/) + !(/') <; !(/+/'). Let s > 0 be a real number, and let h be a function ^0 belonging to Jf R(G) such that h(x) ^ 1 on the union of the (compact) supports off and/' ((3.18.2) and (4.5.2)). Then it is enough to prove that there exists a compact neighbor- hood V of e in G, such that, for all g e 3C*+ with Supp(^) c= V, we have (14.1.5.4) (/: g) + (/': g) ^ (/ + /': g) + e(h : g). To prove this, put u =/+/' +ie/z, and let v (resp. z/) be the function which coincides with//w (resp./'/w) on Supp(/+/') and is zero on the com- plement of this set. Since at every frontier point x of Supp(/+/') we have /(*) +/'(*) = 0> and therefore f(x) =/'(*) = 0, it follows that v and v' belong to JfR(G) and are ^0. The functions v and v' are therefore uniformly continuous with respect to a left-invariant distance defining the topology of G ((12.9.1) and (3.16.5)), and therefore, for each rj > 0, there exists a compact neighborhood V of e such that \v(s) - v(t)\ <£ Y\ and \v'(s) - v'(t)\ s; i; for all pairs (j, 0 such that s""1/e V. Now let getf% be such that Supp(^) <z V. For each j e G we have t? • j(s)g g (u(j) + jy) • y(%. (This is obviously true at points where j(s)g vanishes, hence at points outside ,sV; and if / e ^V we have v(t) ^ v(s) + rj.) Similarly, we have u7 • y(s)g £ (v'(s) + ri)j(s)g. This being so, let c£ (1 ^ / £ri) be real numbers ^0, and st (1 g i ^ /z) elements of G such that tf 5£ £ ciY(si)9> Then we have i=l n / = vu £ £ c, v • _ i=l £=1 and a similar set of inequalities with/replaced by/'. Consequently (/ : 0) + (/': 0) g £ c£(t?(s,) + i;^) + 2iy) ^ (1 4- 2^)^,., because u 4- u' ^ 1. From the definition of w, and properties (ii), (iii), and (v), we obtain (/: 0) + (f'9) g (1 ^ (/+/' : g) so that, by taking q such that we get (14.1.5.4)all//' e ^f(G, Up) n ^f + and all «, because/i-^In( |/j) is a seminorm.